In this article we consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in n-dimensional compact Riemannian manifolds, for . The system we consider consists of a regularized family of Navier–Stokes equations for the fluid velocity u coupled with a convective Allen–Cahn equation for the order (phase) parameter ϕ. We discretize these equations in time using the implicit Euler scheme and we prove that the discrete attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.
Modelling and simulating the behavior of binary fluid mixtures in various turbulent regimes can be rather challenging [1]. A possible approach is based on the so-called diffuse-interface method (see [1,4,20] and their references). This method consists in introducing an order parameter, accounting for the presence of two species, whose dynamics interacts with the fluid velocity. For incompressible fluids with matched densities, a well-known model consists of the classical Navier–Stokes equation suitably coupled with either a convective Cahn–Hilliard or Allen–Cahn equation (see [3,7,8,10,14,15,22,27,31] cf. also [2,5,17–19,21,24]). Denoting by , , the velocity field and by ϕ the order parameter, where we suppose that ϕ is normalized in such a way that the two pure phases of the fluid are and , respectively, the Cahn–Hiliard–Navier–Stokes and the Allen–Cahn–Navier–Stokes systems can be written in a unified form. Indeed, if additionally we assume that the viscosity of fluid is constant, and temperature differences are negligible, we have
in , where Ω is a bounded domain in , , with a sufficiently smooth boundary Γ, is a parameter related to the thickness of the interface separating the two fluids, and is an external body force. Moreover, the operator has a two-fold definition according to the case (Cahn–Hilliard fluid) or (Allen–Cahn fluid), namely,
where is the mobility of the mixture. The so-called chemical potential μ is obtained, under an appropriate choice of boundary conditions, as a variational derivative of the following free energy functional
where , . Here, the potential F is either a double-well logarithmic-type function
for some , or a polynomial approximation of the type
for some .
The system (1.1)–(1.4) captures basic features of binary fluid behavior. The Allen–Cahn and Cahn–Hilliard formulations have each their own advantages and disadvantages (see, e.g., [13]).
It is well known that direct numerical simulation of the 3D NSE for many physical applications with high Reynolds number flows is intractable even using state-of-the-art numerical methods on the most advanced supercomputers available nowadays. Recently, many applied mathematicians have developed regularized turbulence models for the 3D NSE as an attempt to overcome this simulation barrier. Their aim is to capture the large, energetic eddies without having to compute the smallest dynamically relevant eddies, by instead modelling the effects of small eddies in terms of the large scales in the 3D NSE. Since 1998, many such regularized models have been proposed, tested and investigated from both the numerical and the mathematical point of views. Among these models, one can find the globally well-posed 3D Navier–Stokes-α (NS-α) equations (also known as the viscous Camassa–Holm equations and Lagrangian averaged Navier–Stokes-α model), the 3D Leray-α models, the modified 3D Leray-α models, the simplified 3D Bardina models, the 3D Navier–Stokes–Voight (NSV) equations, and their inviscid counterparts. For instance, it has been observed that computational simulations of the 3D Navier–Stokes-α (NS-α) equations are statistically indistinguishable from the simulations of the Navier–Stokes equations. Furthermore, the 3D Navier–Stokes-α model provide a tremendous computational savings as shown in simulations of both forced and decaying turbulence. Finally, the 3D Navier–Stokes-α model arises from a variational principle in the same fashion as the Navier–Stokes equations. We refrain from giving an exhaustive list of references but we refer the reader to [16] for a complete bibliography and detailed description of the results available for these regularized models.
In this paper we consider the following prototype of initial value problem on a d-dimensional compact Riemannian manifold Ω with or without boundary, when :
where , , M and N are bounded linear operators having certain mapping properties, and where χ is either 0 or 1. All kinds of boundary conditions (i.e., periodic, no-slip, no-flux, Navier boundary conditions, etc.) can be treated and are included in our analysis; they will be incorporated in the weak formulation for the problem (1.9), see Section 2. We introduce three parameters which control the degree of smoothing in the operators , M and N, namely θ, and , while is a differential operator of second order. Thus we will only focus on the case when in (1.5), which is actually the harder case. Some examples of operators , , M and N which satisfy the mapping assumptions we will need in this paper are:
for fixed positive real numbers and for specific choices of the real parameters θ, and . We note that the Korteweg force in (1.9) can be equivalently rewritten in the following form
As in [16], we emphasize that the abstract mapping assumptions we employ are more general, and as a result do not require any specific form of the parametrizations of , M and N. This abstraction allows (1.9) to recover some of the existing models that have been previously studied, as well as to represent a much larger three-parameter family of models that have not been explicitly studied in detail. Table 1 presents some regularization models recovered by (1.9) for particular choices of the operators , M, N and χ.
The system (1.9) was considered in [13], where the authors established existence, stability and regularity results, and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the limit in α-models. Then, they also proved the existence of a global attractor and exponential attractor for the general model, and then established precise conditions under which each trajectory converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.
Some special cases of the model (1.9) with , and with and
Model
NSE–AC
Leray–AC-α
ML–AC-α
SBM–AC
NSV–AC
NS–AC-α
NS–AC-α-like
M
I
S
I
S
S
S
N
I
I
S
S
S
I
I
χ
0
0
0
0
0
1
1
The paper is structured as follows. In Section 2, we introduce the notations and give some basic preliminary results for the operators appearing in the general regularized model. Analogous to the theory for the Navier–Stokes–Allen–Cahn system [11,12,26], in Section 3, we prove the long-time stability of the solution of the general three-parameter family of regularized models. In Section 4, we prove the uniqueness of the solution of the discretised system under a time restriction depending on the initial data and explain the need of the theory of the so-called multi-valued attractors. In Section 5, we recall from [6] the main properties of the multi-valued attractors (see Section 5.1) and then we apply the theory to our model.
Preliminary material
We follow the same framework and notation as in [13] (see also [16]). To this end, let Ω be an n-dimensional smooth compact manifold with or without boundary and equipped with a volume form, and let be a vector bundle over Ω equipped with a Riemannian metric . With denoting the space of smooth sections of E, let be a linear subspace, let be a linear operator, and let be a bilinear map. At this point is conceived to be an arbitrary linear subspace of ; however, later on we will impose restrictions on implicitly through various conditions on certain operators such as . Furthermore, we let be a linear subspace and let be a linear operator satisfying various assumptions below. In order to define the variational setting for the phase-field component we also need to introduce the bilinear operators , , as follows:
Recalling (1.11) and assuming that (for the same of simplicity), the initial data , and forcing term with , consider the following system
on the time interval . Bearing in mind the model (1.9), we are mainly interested in bilinear maps of the form
where M and N are linear operators in that are in some sense regularizing and are relatively flexible, and is a bilinear map fixing the underlying nonlinear structure of the fluid equation. In the following, let be the -orthogonal projector onto . Finally, concerning the derivative f of the function F in (1.6) we will focus mostly on the regular potential case when satisfies , and obeys the following condition
We will study the regularized system (2.2) by extending it to function spaces that have weaker differentiability properties. To this end, we interpret (2.2) in distribution sense, and need to continuously extend , and , and to appropriate smoothness spaces. Namely, we employ the spaces , , which will informally be called Sobolev spaces in the following. The pair of spaces and are equipped with the duality pairing , that is, the continuous extension of the -inner product on . Same applies to the triplet . Moreover, we assume that there are self-adjoint positive operators Λ and , respectively, such that , are isometries for any , and , are compact operators. For arbitrary real s, assume that , , M and N can be continuously extended so that
are bounded operators. Again, we emphasize that the assumptions we will need for , M and N are more general, and do not require this particular form of the parametrization (see (2.6)–(2.8)). We will assume and no a priori sign restrictions on , . We remark that s in (2.5) is assumed to be arbitrary for the purpose of the discussion in this section; of course, it suffices to assume (2.5) for a limited range of s for most of the results in this paper. The canonical norm in the Hilbert spaces and , respectively, will be denoted by the same quantity whenever no further confusion arises, while we will use the notation for the -norm. Furthermore, we assume that and N are both self-adjoint, and coercive in the sense that for ,
with , and , and that
with . We also assume that
Note that if , (2.6) is strictly speaking not coercivity and follows from the boundedness of A, and note also that (2.7) implies the invertibility of N. Hereafter, will denote the norm of the operator , which is defined by:
Table 2 presents corresponding values of the parameters and bilinear maps for specific regularization models. For more details on the above spaces and operators, the interested reader is referred to [13].
Values of the parameters θ, and , and the particular form of the bilinear map for some special cases of the model (2.2)
Model
NSE–AC
Leray–AC-α
ML–AC-α
SBM–AC
NSV–AC
NS–AC-α
NS–AC-α-like
θ
1
1
1
1
0
1
θ
0
1
0
1
1
0
0
0
0
1
1
1
1
Note: The bilinear maps and are defined by .
We also introduce the trilinear forms
We define the Hilbert spaces and by
endowed with the scalar products whose associated norms are
Then our notion of weak solution for problem (2.2) can be formulated as follows.
Let for some , and . Find a pair of functions
such that
for some and , such that , and satisfies
for any , such that a.e. on .
As far as the interpretation of the initial conditions , is concerned, note that properties (2.14)–(2.15) imply that and . Thus, the initial conditions are satisfied in a weak sense.
Uniform stability
In this article we consider a time discretization of (2.2) using the fully implicit Euler scheme,
and prove that the attractors generated by the above system converge to the attractor generated by the continuous system (2.2) as the time-step converges to zero.
Hereafter will denote a generic constant whose further dependence on certain quantities will be specified on occurrence and its value can change even within the same line.
We begin with proving the long-time stability of the solution of the general three-parameter family of regularized models.
Uniform boundedness in
In this section we assume that in (1.1), the forcing and we set .
Recall that and .
In proving the uniform boundedness of , we need first to prove a variant of the maximum principle for . In order to do so, we introduce the following truncation operators (cf. [28]), that associate with the function φ, the functions and , given by
Note that we have , , and (see [9])
We are now able to prove the following variant of the maximum principle for :
Suppose that f satisfies (2.4),,and there existssuch that, for any. Moreover, assume that , for any ,and letbe such thata.e. in Ω. Then for any solution to problem (3.1), we have provided that.
Taking the scalar product of the second equation of (3.1) with in and using (3.3), we obtain
Using the assumption that
we find
and then (3.6) yields
Neglecting some positive terms, we find
and thus
Since , we obtain that for all and almost all . Similarly, we can prove that for all and almost all , and the conclusion of the proposition follows right away. □
For every, we have and there existssuch thatand
Taking the scalar product of the first equation of (3.1) with in and using the relation
the skew property (ii) above, as well as (2.8), we obtain
Taking the scalar product of the second equation of (3.1) with in and using (3.15), we obtain
Adding (3.16) and (3.17) and recalling (2.1), (2.10), as well as the third equation of (3.1), we obtain
Using the Cauchy–Schwarz inequality, Proposition 3.1 and (2.5), we majorize the right-hand side of (3.18) by
Combining (3.18)–(3.20), we obtain
and neglecting some positive terms, we find
Recalling (2.5), we have
and setting
and
relation (3.22) yields
Using recursively (3.26), we find
and (3.12) follows right away. Taking
we obtain (3.13).
Now adding inequalities (3.21) with n from i to m and dropping some terms, we find
which is just (3.14) with m in place of n. This completes the proof of the theorem. □
Ifthenwhereand
From the bound (3.12), we infer that
and using assumption (3.30) on k and the fact that if we obtain
For , the above inequality implies conclusion (3.31) of the corollary. □
Uniform boundedness in
We now seek to obtain uniform bounds for in , similar to those we have already obtained in . In order to do so, we will first use the discrete Gronwall lemma to derive an upper bound on , , for some , and then we will use the discrete uniform Gronwall lemma to obtain an upper bound on , .
We begin with some preliminary inequalities.
Letbe a solution of (3.1), corresponding to the initial condition . Assume the following:
is bounded for someandwith,
for anyand,
and, if ,
, if .
Then for every, we have for some.
Taking the scalar product of the first equation of (3.1) with in , of the second equation of (3.1) with , and adding the resulting relations, we obtain
Using (2.7), (2.8), (3.15), the third equation of (3.1), and the fact that
relation (3.33) gives
Using (2.5), we obtain the following bounds:
Recalling Proposition 3.1 and the fact that we have
Now let
Using assumptions (i) and (ii) of the proposition, and recalling (3.13), we obtain
Recalling (2.1) and (2.10), we have
and we bound the terms on the right-hand side of (3.42) depending on whether or .
Case.
Case.
Gathering relations (3.36)–(3.44), we obtain
from which (3.32) follows right away. □
Let,,and letbe a solution of (3.1), corresponding to the initial condition. Assume that, besides the assumptions of Proposition 1, the following conditions hold:
is bounded;
for any.
Then for every, we have
Taking the scalar product of the first equation of (3.1) with in , of the second equation of (3.1) with , and adding the resulting relations, we obtain
Using (2.6), (3.15) and the third equation of (3.1), we find
Since and a.e. on Ω (by Proposition 3.1), we have
Also,
Using the boundedness of (see (i) in the hypothesis), we have
We bound the nonlinear terms in (3.51) depending on whether or .
Case.
Case.
Gathering relations (3.51)–(3.60) and recalling (3.13), we obtain
from which (3.48) follows right away. The proposition has been proved. □
In what follows, we will make use of the following two lemmas, whose proofs can be found in [23]:
Givenand positive sequences,andsuch thatwe have, for any ,
Given, a positive integer , positive sequences ,andsuch thatand given the boundsfor any, we have,
Let,,and letbe a solution of (3.1), corresponding to the initial condition . Under the assumptions of Proposition 1and Proposition2, letbe arbitrarily fixed. Then there exists , such that for every , we have
Moreover, for any initial data , there exists such thatwhereis given in Corollary3.1.
Using (3.32), relation (3.48) gives
which can be rewritten as
with
We have the following bounds:
In order to derive a bound for , we will apply (the discrete uniform Gronwall) Lemma 2. To do so, we recall that , for (see Corollary 3.1), and we compute the following (for ):
Noting that – defined in Proposition 1 – is now independent of the initial value (it depends only on and ), conclusion (3.66) of Proposition 2 yields
which proves (3.69).
Combining the above bound with (3.75), we obtain conclusion (3.67).
Adding (3.61) with n from i to m and using (3.67) gives (3.68). This completes the proof of the theorem. □
Uniqueness
Using the results obtained in the previous sections, we can now prove the following uniqueness theorem:
Under the assumptions of Propositions1and2, the solution of (3.1), corresponding to the initial condition is unique, provided , for some .
Indeed, let and be two solutions corresponding to the same initial data and let and . Then is a solution of the following system:
which can be rewritten as
Taking the scalar product of the first equation with , of the second equation with in and adding the resulting equations, we obtain (using (2.7)):
Using the Cauchy–Schwarz inequality, we majorize the last term on the right-hand side of (4.3) by
Recalling Proposition 3.1 and the fact that we have
and thus relation (4.4) yields
Using assumptions (i) and (ii) of Proposition 1, and recalling (3.67), we obtain
In order to bound the second and third terms on the right-hand side of (4.3) we consider the cases and separately.
Case.
Case.
Combining relations (4.3)–(4.9), we obtain
Taking
relation (4.12) gives , and therefore the uniqueness of the solution. □
Theorem 4 proves the uniqueness of the solution of system (3.1) under constraint (4.13). The dependance of the time-step on the initial data prevents us from defining a single-valued attractor in the classical sense. This is why we need the theory of the so-called multi-valued attractors, which is described below.
Convergence of attractors
In this section we first recall some results on the theory of the multi-valued attractors (see [6,9,25,26]) and then we apply them to our model.
Attractors for multi-valued mappings
Throughout this subsection, we consider to be a Hilbert space and to be either or .
A one-parameter family of set-valued maps is a multi-valued semigroup if it satisfies the following properties:
(identity in );
, for all .
Moreover, the multi-valued semigroup is said to be closed if is a closed map for every , meaning that if in H and is such that in H, then . (To simplify the notation, from now on we will write in place of .)
The positive orbit of , starting at , is the set
where
For any , the set
is called the ω-limit set of .
A nonempty set is invariant for if
A set is an absorbing set for the multi-valued semigroup if for every bounded set there exists such that
A nonempty set is attracting if for every bounded set we have
where is the Hausdorff semidistance, defined as
A nonempty compact set is said to be the global attractor of if is an invariant attracting set.
The global attractor, if it exists, is necessarily unique. Moreover, it enjoys the following maximality and minimality properties:
if is a bounded invariant set, then ;
if is a closed attracting set, then .
Given a bounded set , the Kuratowski measure of noncompactness of is defined as
If is a bounded set in H, then if and only if is compact.
The following theorem, proved in [6], gives conditions for the existence of a global attractor.
Suppose that the closed multi-valued semigrouppossesses a bounded absorbing setandThenis the global attractor of.
Given a set-valued map , we define a discrete multi-valued semigroup by
and we will denote it by (instead of ).
Given two nonempty sets , we write
The main tool in proving the convergence of the discrete attractors to the continuous attractor is the following theorem, whose proof can be found in [6] (see also [9,29,30]).
Letbe a closed multi-valued semigroup, possessing the global attractor, and for, letbe a family of discrete closed multi-valued semigroups, with global attractor. Assume the following:
[Uniform boundedness]: there exists such that the setis bounded in H;
[Finite time uniform convergence]: there exists such that for any,
Thenwhere dist denotes the Hausdorff semidistance defined in (5.1).
Application: A regularized family of models for homogeneous incompressible two-phase flows
Hereafter, we assume that the forcing g is time-independent. For , we define the multi-valued map as follows: for every ,
Following the same ideas as in [6] (see also [9,25,26]), we have the following theorem:
For any, the multivalued map associated to the implicit Euler scheme (3.1) generates a closed discrete multivalued semigroup.
Let, whereis given in Corollary3.1. Then there exists a constant such that for everyand, there exists such thatThus, the set is a-bounded absorbing set for, for .
Let and let be such that . By Corollary 3.1, there exists such that
Taking , relation (3.14) gives
Arguing by contradiction, we obtain that there exists such that
Applying Theorem 3 with initial data we obtain that there exists , such that
Since is an increasing function of its arguments, relations (5.7) and (5.8) give
This completes the proof of Proposition 3. □
Proposition 3 and Theorem 4 give the existence of the discrete global attractors:
For every, there exists the global attractor of the m-semigroup.
Since the global attractor is the smallest closed attracting set of , Proposition 3 implies
and thus
Relation (5.11) implies the uniform boundedness property required by Theorem 5. To prove the finite time uniform convergence we define, for any function ψ and for any , the following:
With the above notations, the system (3.1) can be rewritten as follows, for :
where
Subtracting (5.14) from (2.2) we obtain
where
We rewrite the above system as
where is given in (5.15) and
and we prove the following properties of and :
Letbe arbitrarily fixed and let, whereis given in Corollary3.1. Assume that and letbe the solution of the numerical scheme (3.1). Then there exist andsuch thatand
Since , Theorem 3 and (5.10) yield
Now let be such that , and let be fixed.
Noting that
we have the following bound
Using the continuity of the form (see assumption (i) of Proposition 2) and recalling (3.67), we have the following bound
To bound the remaining terms, we consider the cases and separately.
Case.
Case.
Relations (5.24)–(5.29) imply
and thus, setting and recalling that , we obtain
which proves (5.20).
Now let be such that , and let be fixed
Proceeding as in (5.26) and (5.28), we have the following bounds:
Gathering relations (5.32)–(5.35), we obtain
and thus, setting and recalling that , we obtain
which proves (5.21). □
We can now prove that condition (H2) of Theorem 5 is satisfied.
(Finite time uniform convergence).
For anywe have
Multiplying the first equation of (5.18) by and integrating we obtain (using (2.8))
Multiplying the second equation of (5.18) by and integrating we find
Adding (5.28) and (5.29) and recalling (2.1), (2.10), as well as the third equation of (3.1), we obtain
We have the following bounds:
Proceeding as in (3.43) and (3.46), we also have
Relations (5.41)–(5.47) give
and recalling (2.7) we obtain
where
and
Applying Gronwall’s lemma, we obtain (since )
As shown in [13], the solution of the continuous problem is uniformly bounded in for all . More precisely, we have
Recalling (5.13) and (3.68) we obtain the following bound
for some constant . Similarly,
and hence
Then relation (5.52), combined with (5.20) and (5.21), yields
and hence
The lemma has been proved. □
Both conditions of Theorem 5 being satisfied (see (5.11) and Proposition 5), we conclude
The family of attractorsconverges, as , to , in the sense that where dist is the Hausdorff semidistance defined in (5.1).
Footnotes
Acknowledgements
The author would like to thank the anonymous referees whose comments help to improve the contain of this article.
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